P and Q run along a circular track in 4 minutes and 7 minutes respectively. They start from the same point at the same time. When Q has completed 12 rounds, how many times will they meet if (i) they run in the same direction and (ii) in the opposite direction respectively?

P run along a circular track in 4 minutes

Q run along a circular track in 7 minutes

Let say Track one round length = R

P speed = R/4

Q speed = R/7

1. they run in same direction

P runs faster so p will meet Q when he has done 1 circular round extra

Let say after T min they meet

=> $$T\times\frac{R}{4} = R + T\times\frac{R}{7}$$

=> $$T\times(\frac{R}{4}-\frac{R}{7}) = R  $$

=> T (3R) = 28R

=> T = $$\frac{28}{3}$$ mins

Every $$\frac{28}{3}$$  they meet once

Q completes 12 rounds in $$12\times7 = 84$$ mins

Number of times they meet $$= \frac{84}{\frac{28}{3}} = 9$$

9 Times

2. in opposite direction

They Meet together when they have have completed one ciruclar round together

=> $$T\times\frac{R}{4} = R -T\times\frac{R}{7}$$

=> T =$$\frac{28}{11}$$

Every $$\frac{28}{11}$$ mins they meet

Q completes 12 rounds in $$12\times7 = 84$$ mins

Number of times they meet $$= \frac{84}{\frac{28}{11}} = 33$$

33 Times